MG205: Econometrics Theory and Applications

Empirical Exercise 4: Income Inequality

Jose Ignacio Gonzalez-Rojas
j.i.gonzalez-rojas@lse.ac.uk

London School of Economics and Political Science

January 19, 2026

Today: From Association to Causation

Agenda

Today’s Roadmap

  1. Inequality May Cause Social Problems
  2. Statistical Significance Requires Interpretation
  3. Association Does Not Imply Causation

Term 1 Recap

  • Identification: Can we learn causal parameters from data?
  • Estimation: OLS properties (unbiasedness, consistency)
  • Inference: Hypothesis testing, confidence intervals, p-values

Today: Apply these tools to a real policy debate

Inequality May Cause Social Problems

Redistribution Could Solve Multiple Problems at Once

The Policy Stakes

Wilkinson and Pickett (2009)

The Spirit Level: Why Equality is Better for Everyone

  • Income inequality causes social problems through psychological mechanisms:
    • Status anxiety increases stress
    • Social trust deteriorates
    • Health outcomes worsen

Policy Implications

If true, redistribution policies would:

  • Improve health outcomes
  • Reduce crime and imprisonment
  • Increase educational achievement
  • Build social trust

One policy lever, multiple benefits

Income Inequality Varies Twofold Across Rich Countries

From Japan to Singapore

Inequality Associates with Worse Social Outcomes

Patterns Across Multiple Variables

Most Relationships Are Statistically Significant

Regressions of each outcome on inequality

Outcome Coefficient p-value N
Trust -6.54 0.000 23
Mental Illness +3.50 0.004 12
Imprisonment +0.29 0.001 23
Child Conflict +0.28 0.006 19
Teenage Births +5.86 0.003 21
Drug Use Index +0.34 0.009 22
Outcome Coefficient p-value N
Obesity +2.69 0.020 21
Maths/Literacy -8.09 0.042 21
Life Expectancy -0.31 0.084 23
Infant Mortality +0.26 0.263 23
Homicides +1.84 0.385 23

Statistical Significance Requires Interpretation

Under the Null, This Pattern Would Be Extremely Unlikely

The Logic of Hypothesis Testing

The Hypothesis Test

\[H_0: \beta_1 = 0 \quad \text{(no relationship)}\]

If inequality truly had no effect, how likely would we observe coefficients this large?

Under \(H_0\), with probability \(p\) an outcome this extreme would occur.

But statistical significance is not causal evidence

Large Effects Overcome Small Sample Limitations

Wilkinson and Pickett are correct: the underlying relationships must be powerful.

\[\text{var}(\hat{\beta}) = \frac{\sigma^2}{n}\cdot\frac{1}{\text{var}(x_{i})}\]

Small Sample Challenges

  • Small \(n\) implies large critical value \(c\)
  • Small denominator implies large SE
  • Both make rejection harder

Yet We Still Reject

  • With only 12-23 observations
  • We still find significance
  • The effect must be large

Critics Accused WP of Cherry-Picking Variables

More inequality, more divorce and smoking

Divorce as % of marriages

Smoking prevalence

External Indices Provide Scientific Credibility

UNICEF vs. Author-Created Measures

The Cherry-Picking Critique

  • Authors could select only variables that support their thesis
  • P-hacking:
    • Running many tests
    • Reporting only significant ones
  • Fabricating composite indices to get desired results

The Defense

  • UNICEF child well-being index: externally created
  • Authors cannot manipulate its construction
  • Still shows strong correlation with inequality

Pre-existing indices provide credibility against cherry-picking

Several Relationships Are Driven by Outlying Countries

Sensitivity Analysis

Some relationships may be driven by extreme observations:

  • USA: highest inequality, worst outcomes
  • Japan: lowest inequality, best outcomes

Robustness check

  • Remove outliers and re-estimate
    • If coefficients change dramatically, results are fragile.

What Changes?

Outcome Robust?
Trust Yes
Mental Illness Sensitive
Imprisonment Sensitive to USA
Obesity Yes

Always check sensitivity to outliers

Association Does Not Imply Causation

Omitted Variables Could Explain the Entire Pattern

Culture as a Potential Confounder

The Omitted Variable Problem

  • Cultural factors might cause both:
    • Higher inequality (less redistribution)
    • More social problems (individualism)
  • If so, the correlation is spurious.

Inequality and social problems share a common cause, but inequality does not cause social problems.

Confounder DAG

Control for confounders to isolate causal effect

Never Control for Variables in the Causal Chain

The Mediator Trap

Stress is part of the causal pathway

  • Controlling for stress would block the effect we want to measure.
  • This is the opposite of what we do with confounders.

Confounders and Mediators Require Opposite Treatment

A Critical Distinction

Confounder: CONTROL

  • Opens a backdoor path
  • Creates spurious correlation
  • Solution: Control for Confounder

Mediator: DO NOT CONTROL

  • Part of causal mechanism
  • Blocking removes true effect
  • Solution: Leave Mediator out

US States Offer Different Trade-offs

Alternative Research Designs

Cross-Country Analysis

  • Advantages:
    • Large variation in inequality
    • Policy-relevant comparisons
  • Disadvantages:
    • Only 20-25 observations
    • Cultural confounding
    • Different institutions

US States Analysis

  • Advantages:
    • 50 observations
    • Common institutions
    • Similar culture
  • Disadvantages:
    • Less variation in inequality
    • State-specific confounders
    • Internal migration

Different designs answer similar questions with different assumptions

Four Key Lessons from the Inequality Debate

What to Remember

  1. Large effects can overcome small sample limitations
    • Variance formula explains why significance is possible
  2. External indices provide credibility against cherry-picking
    • UNICEF index cannot be manipulated by researchers
  3. Omitted confounders threaten causal interpretation
    • Culture could explain both inequality and outcomes
  4. Never control for mediators in the causal chain
    • Stress is part of the mechanism, not a confounder

Next Week: Beauty

Does beauty affect earnings in the labor market?

Empirical exercise 5

  • Hamermesh and Biddle (1994)
  • Beauty premium estimates
  • Experimental design